edwin has been saving money from his part - time job at bensons burger shack. he put $932 into a new high…

edwin has been saving money from his part - time job at bensons burger shack. he put $932 into a new high - yield savings account that earns 2% interest annually. write an exponential equation in the form y = a(b)^x that can model the amount of money in edwins account, y, x years after starting the account. use whole numbers, decimals, or simplified fractions for the values of a and b. y = if edwin makes no other deposits or withdrawals, after how many years will his account have more than $1,000? years

edwin has been saving money from his part - time job at bensons burger shack. he put $932 into a new high - yield savings account that earns 2% interest annually. write an exponential equation in the form y = a(b)^x that can model the amount of money in edwins account, y, x years after starting the account. use whole numbers, decimals, or simplified fractions for the values of a and b. y = if edwin makes no other deposits or withdrawals, after how many years will his account have more than $1,000? years

Answer

Explanation:

Step1: Identify the initial - value and growth factor

The initial amount of money $a$ that Edwin puts into the account is $a = 932$. The interest rate is $2%=0.02$. The growth factor $b$ for an exponential - growth model of compound interest is $b=1 + r$, where $r$ is the interest rate. So, $b = 1+0.02=1.02$. The exponential equation is $y=a(b)^x$, substituting the values of $a$ and $b$ we get $y = 932(1.02)^x$.

Step2: Solve for $x$ when $y>1000$

Set up the inequality $932(1.02)^x>1000$. First, divide both sides of the inequality by 932: $(1.02)^x>\frac{1000}{932}\approx1.073$. Take the natural logarithm of both sides: $\ln(1.02^x)>\ln(1.073)$. Using the property of logarithms $\ln(a^b)=b\ln(a)$, we have $x\ln(1.02)>\ln(1.073)$. Then, solve for $x$: $x>\frac{\ln(1.073)}{\ln(1.02)}$. Calculate $\frac{\ln(1.073)}{\ln(1.02)}\approx\frac{0.0704}{0.0198}\approx3.55$. Since $x$ represents the number of years and it must be a whole number, when $x = 4$, the account will have more than $$1000$.

Answer:

$y = 932(1.02)^x$ 4